# diametrically opposed

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Simplified Maximum Likelihood Detecting Algorithm Based on Antipodal Point
Let C = [C.sub.1] [union] [C.sub.2] = [{[X.sub.b]}.sup.B/2.sub.b=1] [union] [{[[bar.X].sub.b]}.sup.B/2.sub.b=1+B/2] be an antipodal constellation defined on [G.sub.T,M], x [epsilon] [C.sub.1] = [{[X.sub.b]}.sup.B/2.sub.b=1] be a T x M transmitted signal matrix, and [bar.X] [member of] [C.sub.2] = [{[bar.X].sub.b]}.sup.B.sub.b=1+B/2] be an antipodal point of X.
[C.sup.2.sub.AO], given a constellation point [X.sub.k,l] [member of] [C.sub.1.sub.AO], then its antipodal point is [X.sub.(k+P/2)(modP),(1+P/2)(modP) [member of] [C.sup.2.sub.AO].
Knowing that the shape of the transmitted signal with normalizing factor of 1/2 is a T x M unitary matrix [bar.X] = [[[E.sup.*][U.sup.*]].sup.*] in [C.sub.AO] and its antipodal point is [bar.X] = [[[E.sup.*][[bar.U].sup.*]].sup.*], if E is invariable via transmission, then the received signal matrix is Y = [[[E.sup.*][V.sup.*]].sup.*], where
In Section 3, we build a framework of USTM constellation based on the antipodal points. The optimal packing method of searching the orthogonal unitary matrices over Grassmannian manifold and the corresponding searching algorithm are investigated.
A pair of antipodal points are defined as two points with the furthest distance on a sphere.
Then the constellation set C = [{[[PHI].sub.b]}.sup.B-1.sub.b=0] is called a framework of the full diversity USTM constellation based on antipodal points on [G.sub.T,M].
for k = 0,1,2,3, {[[OMEGA].sub.k] contains eight points and its antipodal points set, denoted as {[[bar.OMEGA]].sub.k]}, has the following form:
As X, [bar.X] [member of] C [subset] [G.sub.T,M] are a pair of antipodal points, according to Lemma 1, there are [X[bar.X]] = [Q.sub.T] and [[X.sup.*][[bar.X].sup.*]] * = [Q.sup.*.sub.T], where [Q.sub.T] is a T x T unitary matrix.

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